A110090 -id:A110090 - cp's OEIS frontend

A110091 Denominators of sequence of rationals defined by r(n) = n for n<=1 and for n>1: r(n) = (sum of denominators of r(n-1) and r(n-2))/(sum of numerators of r(n-1) and r(n-2)).

1, 1, 1, 3, 1, 3, 5, 1, 5, 7, 1, 7, 9, 1, 9, 11, 1, 11, 13, 1, 13, 15, 1, 15, 17, 1, 17, 19, 1, 19, 21, 1, 21, 23, 1, 23, 25, 1, 25, 27, 1, 27, 29, 1, 29, 31, 1, 31, 33, 1, 33, 35, 1, 35, 37, 1, 37, 39, 1, 39, 41, 1, 41, 43, 1, 43, 45, 1, 45, 47, 1, 47, 49, 1, 49, 51, 1, 51, 53, 1, 53, 55

Offset: 0

Reinhard Zumkeller, Jul 14 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A110090 (numerators).

a[n_]:= a[n]= If[Mod[n, 3]==0, 2*Floor[n/3] +1, If[Mod[n, 3]==1, 1, 2*Floor[n/3] +1]]; Table[a[n], {n, 0, 100}] (* G. C. Greubel, Jun 16 2021 *)

def A110091(n):
    if (n%3==0): return 2*(n//3) + 1
    elif (n%3==1): return 1
    else: return 2*(n//3) +1
[A110091(n) for n in (0..100)] # G. C. Greubel, Jun 16 2021

a(3*k) = 2*k+1, a(3*k+1) = 1, a(3*k+2) = 2*k+1.
r(n) = A110090(n) / a(n).
a(n) = (A110090(n-1) + A110090(n-2))/GCD(a(n-1) + a(n-2), A110090(n-1) + A110090(n-2)).

A138189 Sequence built on pattern (1,-even,-even) beginning 1,-2,-2.

Original entry on oeis.org

1, -2, -2, 1, -4, -4, 1, -6, -6, 1, -8, -8, 1, -10, -10, 1, -12, -12, 1, -14, -14, 1, -16, -16, 1, -18, -18, 1, -20, -20, 1, -22, -22, 1, -24, -24, 1, -26, -26, 1, -28, -28, 1, -30, -30, 1, -32, -32, 1, -34, -34, 1, -36, -36, 1, -38, -38, 1, -40, -40

Offset: 0

Paul Barry, Mar 04 2008

Partial sums of A138188.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A110090, A138188.

b:= [n le 2 select 0 else Abs(2*Self(n-1) -Self(n-2)) -Self(n-1)-1: n in [1..120]];
A138189:= func< n | -b[n+3] >;
[A138189(n): n in [0..100]]; // G. C. Greubel, Jun 16 2021

Join[{1},Riffle[Flatten[{-2#,-2#}&/@Range[25]],1,3]] (* Harvey P. Dale, Nov 02 2011 *)

@CachedFunction
def A138189(n):
    if (n%3==0): return 1
    elif (n%3==1): return -2*(n//3 +1)
    else: return -2*(n//3 +1)
[A138189(n) for n in (0..100)] # G. C. Greubel, Jun 16 2021

G.f.: (1 -2*x -2*x^2 -x^3)/(1 -2*x^3 +x^6).
From G. C. Greubel, Jun 16 2021: (Start)
a(n) = -b(n+3), where b(n) = abs(2*b(n-1) - b(n-2)) - b(n-1) - 1 and b(1) = b(2) = 0.
a(n) = 1 if (n mod 3) = 0, -2*(floor(n/3) + 1) if (n mod 3) = 1 or (n mod 3) = 2. (End)

cp's OEIS Frontend

A110091 Denominators of sequence of rationals defined by r(n) = n for n<=1 and for n>1: r(n) = (sum of denominators of r(n-1) and r(n-2))/(sum of numerators of r(n-1) and r(n-2)).

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